Questions: What would the speed of each particle be if it had the same wavelength as a photon of yellow light (λ=575.0 nm) ? proton (mass =1.673 x 10^-24 g) speed: □ m / s neutron (mass =1.675 x 10^-24 g) speed: □ m/s electron (mass =9.109 x 10^-28 g) speed: □ m / s alpha particle (mass =6.645 x 10^-24 g) speed: □ m / s

Solution Steps

We need to find the speed of different particles (proton, neutron, electron, and alpha particle) if they have the same wavelength as a photon of yellow light with a wavelength of \( \lambda = 575.0 \, \text{nm} \).

The de Broglie wavelength formula relates the wavelength \( \lambda \) of a particle to its momentum \( p \): \[ \lambda = \frac{h}{p} \] where \( h \) is Planck's constant (\( h = 6.626 \times 10^{-34} \, \text{m}^2 \text{kg/s} \)).

The momentum \( p \) of a particle is given by: \[ p = mv \] where \( m \) is the mass of the particle and \( v \) is its velocity.

Rearrange the de Broglie wavelength formula to solve for velocity \( v \): \[ v = \frac{h}{m\lambda} \]

Substitute the given masses and wavelength into the formula to find the velocity for each particle.

• Mass \( m = 1.673 \times 10^{-24} \, \text{g} = 1.673 \times 10^{-27} \, \text{kg} \)
• Wavelength \( \lambda = 575.0 \, \text{nm} = 575.0 \times 10^{-9} \, \text{m} \)

\[ v_{\text{proton}} = \frac{6.626 \times 10^{-34}}{1.673 \times 10^{-27} \times 575.0 \times 10^{-9}} = 6.914 \times 10^{-3} \, \text{m/s} \]

• Mass \( m = 1.675 \times 10^{-24} \, \text{g} = 1.675 \times 10^{-27} \, \text{kg} \)

\[ v_{\text{neutron}} = \frac{6.626 \times 10^{-34}}{1.675 \times 10^{-27} \times 575.0 \times 10^{-9}} = 6.901 \times 10^{-3} \, \text{m/s} \]

• Mass \( m = 9.109 \times 10^{-28} \, \text{g} = 9.109 \times 10^{-31} \, \text{kg} \)

\[ v_{\text{electron}} = \frac{6.626 \times 10^{-34}}{9.109 \times 10^{-31} \times 575.0 \times 10^{-9}} = 1.263 \times 10^{3} \, \text{m/s} \]

• Mass \( m = 6.645 \times 10^{-24} \, \text{g} = 6.645 \times 10^{-27} \, \text{kg} \)

\[ v_{\text{alpha}} = \frac{6.626 \times 10^{-34}}{6.645 \times 10^{-27} \times 575.0 \times 10^{-9}} = 1.732 \times 10^{-3} \, \text{m/s} \]

Final Answer